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Full text of "Treatise On Analysis Vol-Ii"

6 NEGLIGIBLE FUNCTIONS AND SETS 119 oo (13.6.1) If(fn) is a sequence of negligible functions g:0, then £/„ is negligible. This follows immediately from (13.5.8). A subset N of X is said to be negligible (with respect to ju) or ^-negligible if its characteristic function <pN (12.7) is /^-negligible. Clearly any subset of a negligible set is negligible. (13.6.2) A denumerable union of negligible sets is negligible. For if (Nfc) is a sequence of negligible sets and N = (J Nk , we have k = sup <pNk ^ £ cpNk, and the result follows from (13.6.1). For example, with respect to Lebesgue measure A, every set {t} consisting of a single real number is negligible. For if e > 0 there exists a function /e «2f R(R) with values in [0, 1] which is equal to 1 at the point t and vanishes on the complement of the interval [f — e, / + e], and therefore A*(«p{r)) £ A(/) g 28. Hence, by (13.6.2), it follows that every denumerable subset of R (for example, the set Q of rational numbers) is negligible with respect to Lebesgue measure. One can also give examples of nondenumerable sets in R which are negligible with respect to Lebesgue measure (Section 13.8, Problem 4). A property P(;t) of the points of X is said to be true almost everywhere (with respect to IJL) if the complement of the set of points for which P(x) is true is ju-negligible. (13.6.3) A mapping f: X-^R is negligible if and only if it is zero almost everywhere. Let N be the set of all jceX such that \f(x)\ > 0. Then we have <pN g sup n\f\9 and |/| :g sup «<pN, hence the result follows from (13.5.7). n n (1 3.6.4) If a mapping f : X -» R is such that /^*(/) < + oo (resp. ju*(/) > - oo)> thenf(x) < + oo (resp./(x) > — oo) almost everywhere. It is enough to prove the assertion relating to the upper integral. By hypothesis, there exists a function h e */ such that/^ h and /u*(A) < +00. We may therefore limit ourselves to the case where /e »/, and since there is and form an